A PHYSICS PUZZLE FROM MY JEE PREP.

Hey guys! It has been more than a year since I came out of the jee advanced exam centre looking pretty exhausted. A 6 hour marathon comprising of some mind-boggling physics, chemistry and math questions had completely drained me. But the feeling was different than the feeling of completing another 6 hour mock exam. 

It was Relief. 

It was as if someone had lifted a big boulder off my shoulder.

Many people ask - How was your jee prep? Was it too stressful? 

Like most JEE aspirants, I usually look at them as if they have asked the most obvious question in the world. JEE Advanced is widely regarded as one of the toughest undergraduate entrance examinations. Naturally, it demands long hours of study, discipline, consistency, and determination. It is certainly not a piece of cake!

The only thing which kept many of us going was seeing our hard work pay off. Once you have mastered a chapter in a subject, the pleasure of acing every question in your exam is the best feeling. 

Sometimes, people get so passionate about a subject that they start creating problems of their own. I was one of them. I loved physics (which was sadly my let down subject in almost all my exams.). I loved it so much that sometimes I created physics puzzles  in my head and tried to solve it. 

Once I created a problem which I would like to share with you guys. 

Problem (From AIR 2536):

A hollow cylindrical vessel of mass $M$ and uniform cross-sectional area $A$ floats vertically in a large reservoir containing a liquid of density $\rho_l$. Initially, the cylinder is in stable static equilibrium and is immersed to a depth $x_0$.

A thin elastic balloon of negligible membrane mass contains a liquid of density $\rho$ and volume $V$. The balloon is released from rest at a height $h$ directly above the top face of the cylinder and falls vertically.

Upon impact with the cylinder, the balloon transfers mechanical energy to the cylinder–reservoir system, causing the cylinder to move downward. The cylinder remains vertical throughout the motion. Assume that the loss in mechanical energy of the balloon during the collision is transferred entirely to the cylinder–reservoir system, and all dissipative effects (viscosity, turbulence, sound, heat loss, etc.) are negligible.

After the collision, the balloon rebounds vertically upward and reaches a maximum height of $h/2$ above the cylinder.

The cylinder subsequently reaches a maximum additional downward displacement $d$ below its original equilibrium position.

At the exact instant the cylinder reaches this maximum displacement, the balloon ruptures. Exactly $20\%$of the liquid initially contained in the balloon is captured inside the hollow cylinder, while the remaining liquid drains into the reservoir without affecting the subsequent motion.

After all transient oscillations have died out, the cylinder is observed to remain in static equilibrium at an immersion depth of

$$x_0+d.$$Determine an analytical expression for the initial liquid volume 

$V$ contained in the balloon in terms of the relevant parameters of the system.

I found this problem in my laptop quite recently. If you guys are interested, I would solve it and share the solution. 

Good luck!